Demailly's conjecture on Waldschmidt constants for sufficiently many very general points in ${\mathbb{P}}^n$

Abstract

Let $Z$ be a finite set of $s$ points in the projective space ${\mathbb{P}}^n$ over an algebraically closed field $F$. For each positive integer $m$, let $\hat{I}\pm (mZ)$ denote the smallest degree of nonzero homogeneous polynomials in $F[x_0,\dots ,x_n]$ that vanish to order at least $m$ at every point of $Z$. The Waldschmidt constant $\hat{\hat{I}}\pm (Z)$ of $Z$ is defined by the limit \[ \widehatα(Z)=\lim_{m \to \infty}\frac{α(mZ)}{m}. \] Demailly conjectured that \[ \widehatα(Z)\geq\frac{α(mZ)+n-1}{m+n-1}. \] Recently, Malara, Szemberg, and Szpond established Demailly's conjecture when $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor-2\geq m-1. \] Here we improve their result and show that Demailly's conjecture holds if $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor-2\ge \frac{2\varepsilon}{n-1}(m-1), \] where $0\le \epsilon <1$ is the fractional part of $\sqrt[n]{s}$. In particular, for $s$ very general points where $\sqrt[n]{s}\in \mathbb{N}$ (namely $\epsilon =0$), Demailly's conjecture holds for all $m\in \mathbb{N}$. We also show that Demailly's conjecture holds if $Z$ is very general and \[ s\ge\max\{n+7,2^n\}, \] assuming the Nagata-Iarrobino conjecture $\hat{\hat{I}}\pm (Z)\ge \sqrt[n]{s}$.